If mental illness does not equal insanity (in a trial), what does?

I’m watching the trial of Iowa vs. Becker on court TV. In the trial a man who was diagnosed with paranoid schizophrenia killed his coach just days after getting released from a mental institution. He told police that he killed the coach because the coach was turning him and the other students into “fish, animals, and dead people” and that the coach was a devil, along with other things that are obviously coming from someone with a mental illness (saying if you squeezed his right hand then the devil wouldn’t be so powerful, the coach was keeping the devil in his nostrils, etc).

The defense is claiming not guilty due to insanity. However, the prosecution argued that “mental illness does not equal insanity”.

If such a debilitating illness like paranoid schizophrenia doesn’t count as insanity in the court, what does? I’m just trying to understand, as I’ve taken an interest in watching these trials. My only guess is that it might be considered insanity if, say, someone kills your child right in front of you, so you temporarily lose your sense of judgment and kill the person who killed your child?

I dunno. Does anyone have any insight?